Accurate models for the threshold voltage and subthreshold swing in nanoscale MOSFETs should be based forom a 2D electrostatic analysis.
In doped devices, the carrier concentration is usually neglected in such electrostatic analysis, that is carried out in the subthreshold region, where the doping concentration is much higher than the carrier concentration.
However, in the case of undoped DG MOSFETs, the carrier concentration may be not negligible for that electrostatic analysis, in particular in the threshold regime.
A recent paper was presenting the first complete threshold voltage and subthreshold swing models for nanoscale DG MOSFETs, by considering the carrier concentration in Poisson's equation:
El Hamid, H. A.; Roig Guitart, J.; Iniguez, B., "Two-Dimensional Analytical Threshold Voltage and Subthreshold Swing Models of Undoped Symmetric Double-Gate MOSFETs," IEEE Trans. on Electron Devices, vol. 54, no. 6, pp. 1402-1408, June 2007
This model includes the threshold voltage roll-off, the DIBL and the subthreshold swing degradation.
This paper improves the model developed by Qian Chen et al., also published in IEEE Transactions on Electron Devices, in which the carrier concentration was considered, but the model development was done only at low drain voltage; the DIBL and the subthreshold swing degradation with the drain voltage were ignored:
Q. Chen, E. M. Harrell, J. D. Meindl, "A physical short-channel threshold voltage model for undoped symmetric double-gate MOSFETs," IEEE Transactions on Electron Devices, vol. 50, no. 7, pp. 1631-1637, July 2003